The aim of these lectures is to introduce basic processes responsible for cooling
of neutron stars and to show how to calculate the neutrino production rate in
dense strongly interacting nuclear medium. The formalism is presented that
treats on equal footing one-nucleon and multiple-nucleon processes and reactions
with virtual bosonic modes and condensates. We demonstrate that neutrino
emission from dense hadronic component in neutron stars is subject of strong
modiﬁcations due to collective eﬀects in the nuclear matter. With the most
important in-medium processes incorporated in the cooling code an overall
agreement with available soft X ray data can be easily achieved. With these
ﬁndings the so-called “standard” and “non-standard” cooling scenarios are
replaced by one general “nuclear medium cooling scenario” which relates slow and
rapid neutron star coolings to the star masses (interior densities).

The lectures are split in four parts.

Part I: After short introduction to the neutron star cooling problem we show how
to calculate neutrino reaction rates of the most eﬃcient one-nucleon and
two-nucleon processes. No medium eﬀects are taken into account in this instance.
The eﬀects of a possible nucleon pairing are discussed. We demonstrate that the
data on neutron star cooling cannot be described without inclusion of medium
eﬀects. It motivates an assumption that masses of the neutron stars are diﬀerent
and that neutrino reaction rates should be strongly density dependent.

Part II: We introduce the Green’s function diagram technique for systems in and
out of equilibrium and the optical theorem formalism. The latter allows to
perform calculations of production rates with full Green’s functions including
all oﬀ-mass-shell eﬀects. We demonstrate how this formalism works within the
quasiparticle approximation.

Part III: The basic concepts of the nuclear Fermi
liquid approach are introduced. We show how strong interaction eﬀects can be
included within the Green’s function formalism. Softening of the pion mode with
an baryon density increase is explicitly incorporated. We show examples of
inconsistencies in calculations without inclusion of medium eﬀects. Then we
demonstrate calculations of diﬀerent reaction rates in non-superﬂuid nuclear
matter with taking into account medium eﬀects. Many new reaction channels are
open up in the medium and should be analyzed.

Part IV: We discuss the neutrino production reactions in superﬂuid nuclear
systems. The reaction rates of processes associated with the pair breaking and
formation are calculated. Special attention is focused on the gauge invariance
and the exact fulﬁllment of the Ward identities for the vector current. Finally
we present comparison of calculations of neutron star cooling performed within
nuclear medium cooling scenario with the available data.

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